Optimal. Leaf size=160 \[ \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{3/2}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 \sqrt {2+3 x}}-\frac {5594 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15435}-\frac {1196 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15435} \]
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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164,
114, 120} \begin {gather*} -\frac {1196 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15435}-\frac {5594 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15435}+\frac {5594 \sqrt {1-2 x} \sqrt {5 x+3}}{15435 \sqrt {3 x+2}}-\frac {404 \sqrt {1-2 x} \sqrt {5 x+3}}{2205 (3 x+2)^{3/2}}+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}-\frac {2}{105} \int \frac {-248-\frac {845 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{3/2}}-\frac {4 \int \frac {-\frac {2279}{4}-505 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{2205}\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{3/2}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 \sqrt {2+3 x}}-\frac {8 \int \frac {-2920-\frac {13985 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15435}\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{3/2}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 \sqrt {2+3 x}}+\frac {5594 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{15435}+\frac {6578 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15435}\\ &=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{3/2}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{15435 \sqrt {2+3 x}}-\frac {5594 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15435}-\frac {1196 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15435}\\ \end {align*}
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Mathematica [A]
time = 4.62, size = 99, normalized size = 0.62 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (8507+29322 x+25173 x^2\right )}{(2+3 x)^{5/2}}+\sqrt {2} \left (2797 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+7070 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{46305} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs.
\(2(116)=232\).
time = 0.10, size = 308, normalized size = 1.92
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2835 \left (\frac {2}{3}+x \right )^{3}}-\frac {404 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19845 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {11188}{3087} x^{2}-\frac {5594}{15435} x +\frac {5594}{5145}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {4672 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{64827 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5594 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{64827 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(249\) |
default | \(-\frac {2 \left (88803 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-25173 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+118404 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-33564 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+39468 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-11188 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-755190 x^{4}-955179 x^{3}-116619 x^{2}+238377 x +76563\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{46305 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.20, size = 50, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (25173 \, x^{2} + 29322 \, x + 8507\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{15435 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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